Evidence for the Poisson/Gaudin--Mehta phase transition for band matrices on global scales

Abstract

We prove that the Poisson/Gaudin--Mehta phase transition conjectured to occur when the bandwidth of an N ⨯ N symmetric band matrix grows like b = √N is naturally observable in the rate of convergence of the level density to the Wigner semi-circle law. Specifically, we show for periodic and non-periodic band matrices the rate of convergence of the fourth moment of the level density is independent of the boundary conditions in the localised regime b << √N with a rate of O(1/b) for both cases, whereas in the delocalised regime b >> √N where boundary effects become important, the rate of convergence for the two ensembles differ significantly, slowing to O(b/N) for non-periodic band matrices. Additionally, we examine the case of thick non-periodic band matrices b = cN, showing that the fourth moment is maximally deviated from the Wigner semi-circle law when b = 2N/5, and provide numerical evidence that the eigenvector statistics also exhibit critical behaviour at this point.